LECTURE 7: PROPOSITIONAL
LOGIC (1)
Software Engineering
Mike Wooldridge
Lecture 7
Software Engineering
1 What is a Logic?
• When most people say ‘logic’, they mean
either propositional logic or first-order
predicate logic.
• However, the precise definition is quite
broad, and literally hundreds of logics
have been studied by philosophers,
computer scientists and mathematicians.
• Any ‘formal system’ can be considered a
logic if it has:
– a well-defined syntax;
– a well-defined semantics; and
– a well-defined proof-theory.
Mike Wooldridge
1
Lecture 7
Software Engineering
• The syntax of a logic defines the
syntactically acceptable objects of the
language, which are properly called
well-formed formulae (wff). (We shall just
call them formulae.)
• The semantics of a logic associate each
formula with a meaning.
• The proof theory is concerned with
manipulating formulae according to
certain rules.
Mike Wooldridge
2
Lecture 7
Software Engineering
2 Propositional Logic
• The simplest, and most abstract logic we
can study is called propositional logic.
• Definition: A proposition is a statement
that can be either true or false; it must be
one or the other, and it cannot be both.
• EXAMPLES. The following are
propositions:
– the reactor is on;
– the wing-flaps are up;
– John Major is prime minister.
whereas the following are not:
– are you going out somewhere?
– 2+3
Mike Wooldridge
3
Lecture 7
Software Engineering
• It is possible to determine whether any
given statement is a proposition by
prefixing it with:
It is true that . . .
and seeing whether the result makes
grammatical sense.
• We now define atomic propositions.
Intuitively, these are the set of smallest
propositions.
• Definition: An atomic proposition is one
whose truth or falsity does not depend on
the truth or falsity of any other
proposition.
• So all the above propositions are atomic.
Mike Wooldridge
4
Lecture 7
Software Engineering
• Now, rather than write out propositions in
full, we will abbreviate them by using
propositional variables.
• It is standard practice to use the
lower-case roman letters
p, q, r, . . .
to stand for propositions.
• If we do this, we must define what we
mean by writing something like:
Let p be John Major is prime Minister.
• Another alternative is to write something
like reactor is on, so that the interpretation
of the propositional variable becomes
obvious.
Mike Wooldridge
5
Lecture 7
Software Engineering
2.1 The Connectives
• Now, the study of atomic propositions is
pretty boring. We therefore now introduce
a number of connectives which will allow
us to build up complex propositions.
• The connectives we introduce are:
∧
∨
¬
⇒
⇔
and (& or .)
or (| or +)
not (∼)
implies (⊃ or →)
iff
• (Some books use other notations; these are
given in parentheses.)
Mike Wooldridge
6
Lecture 7
Software Engineering
And
• Any two propositions can be combined to
form a third proposition called the
conjunction of the original propositions.
• Definition: If p and q are arbitrary
propositions, then the conjunction of p and
q is written
p∧q
and will be true iff both p and q are true.
Mike Wooldridge
7
Lecture 7
Software Engineering
• We can summarise the operation of ∧ in a
truth table. The idea of a truth table for
some formula is that it describes the
behaviour of a formula under all possible
interpretations of the primitive
propositions the are included in the
formula.
• If there are n different atomic propositions
in some formula, then there are 2n different
lines in the truth table for that formula.
(This is because each proposition can take
one 1 of 2 values — true or false.)
• Let us write T for truth, and F for falsity.
Then the truth table for p ∧ q is:
p
F
F
T
T
Mike Wooldridge
q p∧q
F
F
T
F
F
F
T
T
8
Lecture 7
Software Engineering
Or
• Any two propositions can be combined by
the word ‘or’ to form a third proposition
called the disjunction of the originals.
• Definition: If p and q are arbitrary
propositions, then the disjunction of p and
q is written
p∨q
and will be true iff either p is true, or q is
true, or both p and q are true.
Mike Wooldridge
9
Lecture 7
Software Engineering
• The operation of ∨ is summarised in the
following truth table:
p
F
F
T
T
Mike Wooldridge
q p∨q
F
F
T
T
F
T
T
T
10
Lecture 7
Software Engineering
If. . . Then. . .
• Many statements, particularly in
mathematics, are of the form:
if p is true then q is true.
Another way of saying the same thing is to
write:
p implies q.
• In propositional logic, we have a
connective that combines two propositions
into a new proposition called the
conditional, or implication of the originals,
that attempts to capture the sense of such
a statement.
Mike Wooldridge
11
Lecture 7
Software Engineering
• Definition: If p and q are arbitrary
propositions, then the conditional of p and q
is written
p⇒q
and will be true iff either p is false or q is
true.
• The truth table for ⇒ is:
p q p⇒q
F F
T
F T
T
T F
F
T T
T
Mike Wooldridge
12
Lecture 7
Software Engineering
• The ⇒ operator is the hardest to
understand of the operators we have
considered so far, and yet it is extremely
important.
• If you find it difficult to understand, just
remember that the p ⇒ q means ‘if p is
true, then q is true’.
If p is false, then we don’t care about q, and
by default, make p ⇒ q evaluate to T in
this case.
• Terminology: if φ is the formula p ⇒ q,
then p is the antecedent of φ and q is the
consequent.
Mike Wooldridge
13
Lecture 7
Software Engineering
Iff
• Another common form of statement in
maths is:
p is true if, and only if, q is true.
• The sense of such statements is captured
using the biconditional operator.
• Definition: If p and q are arbitrary
propositions, then the biconditional of p
and q is written:
p⇔q
and will be true iff either:
1. p and q are both true; or
2. p and q are both false.
Mike Wooldridge
14
Lecture 7
Software Engineering
• The truth table for ⇔ is:
p q p⇔q
F F
T
F T
F
T F
F
T T
T
• If p ⇔ q is true, then p and q are said to be
logically equivalent. They will be true under
exactly the same circumstances.
Mike Wooldridge
15
Lecture 7
Software Engineering
Not
• All of the connectives we have considered
so far have been binary: they have taken
two arguments.
• The final connective we consider here is
unary. It only takes one argument.
• Any proposition can be prefixed by the
word ‘not’ to form a second proposition
called the negation of the original.
• Definition: If p is an arbitrary proposition
then the negation of p is written
¬p
and will be true iff p is false.
• Truth table for ¬:
p ¬p
F T
T F
Mike Wooldridge
16
Lecture 7
Software Engineering
Comments
• We can nest complex formulae as deeply as
we want.
• We can use parentheses i.e., ),(, to
disambiguate formulae.
• EXAMPLES. If p, q, r, s and t are atomic
propositions, then all of the following are
formulae:
p∧q⇒r
p ∧ (q ⇒ r)
(p ∧ (q ⇒ r)) ∨ s
((p ∧ (q ⇒ r)) ∨ s) ∧ t
whereas none of the following is:
p∧
p ∧ q)
p¬
Mike Wooldridge
17
Lecture 7
Software Engineering
3 Tautologies & Consistency
• Given a particular formula, can you tell if
it is true or not?
• No — you usually need to know the truth
values of the component atomic
propositions in order to be able to tell
whether a formula is true.
• Definition: A valuation is a function which
assigns a truth value to each primitive
proposition.
• In Modula-2, we might write:
PROCEDURE Val(p : AtomicProp):
BOOLEAN;
• Given a valuation, we can say for any
formula whether it is true or false.
Mike Wooldridge
18
Lecture 7
Software Engineering
• EXAMPLE. Suppose we have a valuation
v, such that:
v(p) = F
v(q) = T
v(r) = F
Then we truth value of (p ∨ q) ⇒ r is
evaluated by:
(v(p) ∨ v(q)) ⇒ v(r)
= (F ∨ T) ⇒ F
=T⇒F
=F
(1)
(2)
(3)
(4)
Line (3) is justified since we know that
F ∨ T = T.
Line (4) is justified since T ⇒ F = F.
If you can’t see this, look at the truth tables
for ∨ and ⇒.
Mike Wooldridge
19
Lecture 7
Software Engineering
• When we consider formulae in terms of
interpretations, it turns out that some have
interesting properties.
• Definition:
1. A formula is a tautology iff it is true
under every valuation;
2. A formula is consistent iff it is true
under at least one valuation;
3. A formula is inconsistent iff it is not
made true under any valuation.
• Now, each line in the truth table of a
formula correponds to a valuation.
• So, we can use truth tables to determine
whether or not formulae are tautologies.
Mike Wooldridge
20